Brilli the ant has a block of cheese that is in the shape of an equilateral triangle. She is hosting a party, and has invited her two-legged house mates. She insists that all of her guests get a piece of cheese that is in the shape of an isosceles triangle. What is the maximal number N such that she cannot split the equilateral triangle block of cheese into N non-degenerate isosceles triangles?

She cannot subdivide the equilateral triangle into 2 isosceles triangles. This is the smallest N which is impossible.

She can subdivide it into 3 congruent isosceles triangles by joining the centroid with each of the vertices. Each triangle is then 30-30-120 as angles.

Each of the 30-30-120 triangles can in turn be subdivided into 3 isosceles triangles with angles 60-60-60, and 30-30-120, thus adding two more triangles.

So the next possible N are then 5, 7, 9.

Since each of the smaller 30-30-120 triangles can again be subdivided, indefinitely, we conclude that the original equilateral triangle can be subdivided into odd N above 1. Even N are therefore not possible. Thus there is no maximum number N, since N->∞, as long as N=2k, where k∈integers.

MathMate, I would disagree with you because 4 triangles are possible. We can break up the equilateral triangle to 4 different equilateral triangles.

To solve this problem, let's consider the properties of the equilateral triangle and the isosceles triangle.

An equilateral triangle has three equal sides and three equal angles of 60 degrees each. An isosceles triangle, on the other hand, has two equal sides and two equal angles.

Let's analyze the possible ways to split the equilateral triangle into isosceles triangles:

1. Splitting the equilateral triangle into two isosceles triangles: In this case, we can see that the equilateral triangle can be split into two equal isosceles triangles along the altitude. Each of these triangles will have a 60-degree base angle and two equal sides.

2. Splitting the equilateral triangle into three isosceles triangles: In this case, we can split the equilateral triangle into three smaller isosceles triangles by drawing lines from each vertex to the midpoint of the opposite side. Each smaller triangle will have a 30-degree base angle and two equal sides.

3. Splitting the equilateral triangle into four isosceles triangles: We can achieve this by drawing lines from each vertex to the midpoints of the adjacent sides. Each resulting triangle will have a 45-degree base angle and two equal sides.

Now, let's analyze the possibilities for larger numbers of isosceles triangles:

- For five isosceles triangles, we would need to create intersecting lines from the vertices to form smaller triangles. However, this would result in degenerate triangles (triangles with zero area).

- By extending this reasoning, we can see that any odd number of isosceles triangles cannot be formed without degenerate triangles.

Therefore, the maximal number N such that she cannot split the equilateral triangle block of cheese into N non-degenerate isosceles triangles is 4.

To understand this concept visually, you can also draw the equilateral triangle and the lines of the possible splits to better visualize the shape and arrangement of the resulting isosceles triangles.